On 7/7/2014 5:22 AM, Alex Shkotin wrote:
> I like idea that set theory may be over reduced to one prime
> predicate symbol ∈. Maybe it's like a Scheffer stroke for
> propositional calculus :-) In this case category should replace
> algebraic system as a model for theory? (01)
This gets into the issue of whether a single foundation for all
of mathematics is necessary, desirable, useful, or deadening. (02)
Marvin Minsky made a good point, which I believe applies to nearly
every subject: "If you only understand something in one way,
you don't really understand it." (03)
The Sheffer stroke (discovered by Peirce about 30 years earlier --
when he wrote the truth tables for all 16 dyadic Boolean operators)
is an example of an artificial foundation that is rarely used in
knowledge representation. But the nand & nor operators turned
out to be useful in designing computer circuits. (04)
Another artificial example is the Dedekind cut for constructing
a model of the real numbers in terms of the rationals. It's useful
for relating the axioms for real numbers to the axioms for integers.
But mathematicians never use it in any practical applications. (05)
Another example is the use of infinitesimals for the foundations of
calculus. Leibniz used them to discover and explain the techniques
of calculus, and his notation of dx/dy is based on them. Euler,
Lagrange, Gauss, etc., used infinitesimals in all their discoveries.
But they were replaced by epsilons and deltas as a more "rigorous"
foundation in the late 19th c. (06)
However, the techniques for using infinitesimals are conceptually
much simpler than using ε and δ. Epsilons and deltas probably
caused more students to fail calculus than anything else. (07)
Furthermore, physicists and even professional mathematicians continued
to use infinitesimals because they were heuristically far more useful: (08)
From http://plato.stanford.edu/entries/continuity/
> Physicists and engineers, for example, never abandoned their use
> as a heuristic device for the derivation of correct results in the
> application of the calculus to physical problems. Differential
> geometers of the stature of Lie and Cartan relied on their use
> in the formulation of concepts which would later be put on a
> “rigorous” footing. (09)
This article is an good overview of the historical developments
from the ancient Greeks to the latest R & D. (010)
For teaching introductory calculus, I strongly agree with the idea
of starting with infinitesimals -- and only talking about epsilons
and deltas as an *alternative* method, not the primary one. (011)
That's how I learned calculus: I read my father's old calculus book,
which used infinitesimals and only mentioned epsilons and deltas
after the readers had learned to solve problems. It enabled me to
pass the advanced placement test and skip the freshman year of
calculus at MIT. (012)
John (013)
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